Re: Cantor and the binary tree

Virgil said:
> In article <1116959066.601645.283260@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> mueckenh@xxxxxxxxxxxxxxxxx wrote:
>
> > Robin Chapman wrote:
> >
> >
> > > >> of paths always equals that of the nodes + 1. It is simply
> > impossible
> > > >> to assume that one of these numbers becomes uncountably infinite
> > while
> > > >> the other remains countably infinite.
> > >
> > > "becomes"?
> >
> > If you prefer "is", you may use it. That does not matter. It is
> > obviously impossible that the set of paths is uncountable when the set
> > of nodes is countable, because every pair of paths springs off from one
> > node, while one path leads to that node. Try to find an error n the
> > arguing, not in the result.
>
> For finite paths (having a root node and a leaf or terminal node) one is
> there can only be countably many, but it is those paths starting at the
> root which procede through infinitely many nodes in this infinite tree,
> and never have a terminal or leaf node, that are under consideration.
>
> The set of these unbounded paths can mapped bijectively to P(N), whereas
> the set of nodes can be mapped bijectively to N.

Then bijections are foolish at infinity. Regardless of finitude or lack
thereof, the number of paths NEVER surpasses the number of nodes. It is simply
not possible, despite your pseudo-logical system.
> >
> > > One can hardly imagine a simpler mathematical proof. Alas, it's still
> > > beyond the limits of Chaps
>
> As all such false proofs should be.
>

--
Smiles,

Tony
.

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